- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Exponential and trigonometric functions practice question 10
Find every in at which the graph of has a horizontal tangent, and enter the sum of those values.
What is the slope of a horizontal line, and what does that say about at the points you want?
. Setting it to zero means .
means . Where is that true in ?
An exact multiple of pi, or a decimal.
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Horizontal tangents are zeros of the derivative. The derivative is cos x − sin x, which vanishes where sine and cosine are equal.
Working
Dividing by cos x is safe: cos x = 0 would force sin x = 0 too, impossible.
tan x = 1 in the first and third quadrants.
Sum.
Answer
The answer is pi over 4 and 5 pi over 4, whose sum is 3 pi over 2.
Check
Written as √2 sin(x + π/4), sin x + cos x peaks at x = π/4 and bottoms out at x = 5π/4, its only flat points in [0, 2π].
Where this goes wrong
Stopping at x = π/4. The equation tan x = 1 has two solutions per full turn, and 5π/4 is the second.
Leans on
Solving a trigonometric equation, from Functions, graphs and inverses
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